# What's different of power spectrum and correlation function of BAO?

(source)

There is the sentence "the harmonic sequence in the power spectrum correspond to a single peak in the correlation function and that the damping envelope corresponds to the broadening of this peak" in "Eisenstein et al. (2007) APJ_664_675".

Before, I misunderstood that the BAO peak in the correlation function has many peaks but other peaks are so small that we cannot see these except for the 150Mpc peak.

But this paper states "a single peak in the correlation function", so I want to ask: Why are there many peaks in the power spectrum, but only a single peak in the correlation function?

The power spectrum and the correlation function are related by

$$\xi(r) = {1 \over 2\pi^2} \int dk\ k^2 P(k) {\sin(kr) \over kr}$$

They are (sort of) the Fourier transform of each other. It should not surprise, then, that a peak in one becomes a series of oscillations in the other. Just like the fourier transform of a sinusoidal wave is a dirac delta.

Keep also in mind that the first figure in the question is not exactly the power spectrum. As indicated by the y-axis label, a baseline has been subtracted and it is showing only the oscillations over that baseline.

The full power spectrum looks like this one:

• I have a one more question. What does the k means in terms of physics?
– BAO
Commented Oct 11, 2021 at 13:25
• @BAO $k$ is the comoving wavenumber. We are moving from physical space to fourier space. We describe the density function as a sum (integral) of different oscillating components $\sin(kr)$. The component with wavenumber $k$ corresponds to an oscillation with physical wavelength $\lambda = 2\pi a / k$. The bigger $k$, the smaller the oscillation. Commented Oct 11, 2021 at 21:45
• Please, tell me if you'd like a more thorough explanation of the words I used Commented Oct 11, 2021 at 21:50